This is page 23 (the last) of the Cosmic Call
message. An explanation follows.

This page is a series of questions for the recipients of the message.
It is labeled with the glyph , which heretofore
appeared only on page 4 in the context
of solving of algebraic equations. So we might interpret it as
meaning a solution or a desire to solve or understand. I have chosen to translate
it as “wat”.

I find this page irritating in its vagueness and confusion. Its
layout is disorganized. Glyphs are used inconsistent with their uses
elsewhere on the page and elsewhere in the message.
For example, the mysterious glyph
, which
has something to do with the recipients of the message, and which
appeared only on page 21 is used here to
ask about both the recipients themselves and also about their planet.

The questions are arranged in groups. For easy identification, I have
color-coded the groups.

Starting from the upper-left corner, and proceeding counterclockwise, we
have:

Kilograms, meters, and seconds, wat. I would have used the glyphs for
abstract mass, distance, and time,
and ,
since that seems to be closer to the intended meaning.

Alien mathematics, physics, and biology, wat. Note that this asks
specifically about the recipients’ version of the sciences.
None of these three glyphs has been subscripted before. Will the
meaning be clear to the recipients? One also wonders why the message
doesn't express a desire to understand human science, or science
generally. One might argue that it does not make sense to ask the
recipients about the human versions of mathematics and physics. But a
later group expresses a desire to understand males and females, and the
recipients don't know anything about that either.

Aliens wat. Alien [planet] mass, radius, acceleration wat.
The meaning of
shifts here from meaning the recipients themselves to the recipients’
planet. “Acceleration”
is intended to refer to the planet's gravitational acceleration as
on page 14. What if the recipients
don't live on a planet? I suppose they will be familiar with planets
generally and with the fact that we live on a planet, which explained
back on pages 11–13, and will get the idea.

Fucking speed of light, how does it work?

Planck's constant, wat. Universal gravitation constant, wat?

Males and females, wat. Alien people, wat. Age of people,
wat. This group seems to be about our desire to understand ourselves,
except that the third item relates to the aliens. I'm not quite sure
what is going on. Perhaps “males and females” is intended to refer to
the recipients? But the glyphs are not subscripted, and there is no
strong reason to believe that the aliens have the same sexuality.

The glyph
, already used
both to mean the age of the Earth and the typical human lifespan, is
even less clear here. Does it mean we want to understand the reasons
for human life expectancy? Or is it intended to continue the inquiry
from the previous line and is asking about the recipients’ history or
lifespan?

Land, water, and atmosphere of the recipients’ planet, wat.

Energy, force, pressure, power, wat. The usage here is
inconsistent from the first group, which asked not about mass,
distance, and time but about kilograms, meters, and seconds specifically.

Velocity and acceleration, wat. I wonder why these are in a
separate group, instead of being clustered with the previous group or
the first group. I also worry about the equivocation in
acceleration,
which is sometimes used to mean the Earth's gravitational acceleration
and sometimes acceleration generally. We already said we want to
understand
mass ,
!!G!! ,
and the size of the Earth. The Earth's surface gravity can be
straightforwardly calculated from these, so there's nothing else to
understand about that.

Alien planet, wat.
The glyph
has
heretofore been used only to refer to the planet Earth. It does not mean planets
generally, because it was not used in connection with Jupiter
.
Here, however, it
seems to refer to the recipients’ planet.

The universe, wat. HUH???

That was the last page. Thanks for your kind attention.

[ Many thanks to Anna Gundlach, without whose timely email I might not have found the motivation to finish this series. ]

This page discusses properties of the entire universe. It is labeled
with a new glyph,
,
which denotes the universe or the cosmos. On this page I am on
uncertain ground, because I know very little about cosmology. My
explanation here could be completely wrong without my realizing it.

The page contains only five lines of text. In order, they state:

The Friedmann
equation
which is the current model for the expansion of the
universe. This expansion is believed to be uniform everywhere, but
even if it isn't, the recipients are so close by that they will
see exactly the same expansion we do. If they have noticed the expansion,
they may well have come to the same theoretical conclusions about it.
The equation is:

$$H^2 = \frac{8\pi G}3\rho + \frac{\Lambda c^2 }3$$

where !!H!! is the Hubble parameter
(which describes how quickly the universe is expanding),
!!G!! is the universal gravitation constant
(introduced on page 9),
!!\rho!! is the density of the universe
(given on the next line),
and !!\Lambda c^2!! ()
is one of the forms of the
cosmological constant (given on the following line).

The average density
of the universe ,
given as !!2.76\times 10^{-27} \mathrm{kg}
~\mathrm{m}^{-3}!!. The “density” glyph would have been more at home
with the other physics definitions of page
9, but it wasn't needed until now, and
that page was full.

The cosmological constant !!\Lambda!! is about !!10^{-52}
\mathrm{m}^{-2}!!. The related value given here, !!\Lambda c^2!!, is !!1.08\cdot
10^{-35} \mathrm{s}^{-2}!!.

The calculated value of the Hubble parameter !!H!! is given here in
the rather strange form !!\frac1{14000000000}\mathrm{year}^{-1}!!.
The reason it is phrased this way is that (assuming that !!H!! were
constant) !!\frac1H!! would be the age of the universe, approximately
14,000,000,000 years. So this line not only communicates our
estimate for the current value of the Hubble parameter, it
expresses it in units that may make clear our beliefs about the age
of the universe. It is regrettable that this wasn't stated more
explicitly, using the glyph
that was already used for the age of the Earth on page
13. There
was plenty of extra space, so perhaps the senders didn't think of it.

The average temperature of the universe, about 2.736 kelvins. This is based on measurements of the cosmic microwave background radiation, which is the same in every direction, so if the recipients have noticed it at all, they have seen the same CMB that we have.

The next
article will discuss the final page, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page discusses the message itself. It is headed with the glyph
for “physics” .

The
leftmost part of the page has a cartoon of the Yevpatoria RT-70 radio
telescope
that was used to send the message, labeled “Earth” . Coming out the
the telescope is a stylized depiction of a radio wave. Two rulers
measure the radio wave. The smaller one measures a single wavelength,
and is labeled “frequency =
5,010,240,000 Hz ” and “wavelength
=
0.059836 meters ”; these are the
frequency and the wavelength of the radio waves used to send the
message. The longer ruler has the notation “127×127×23”, describing
the format of the message itself, 23 pages of 127×127 bitmaps, and
also “43000 people ”, which I do not
understand at all. Were 43,000 people somehow involved with sending
the message? That seems far too many. Were there 43,000 people in
Yevpatoria in 1999? That seems far too few; the current population is
over 100,000. I am mystified.

At the other end of the radio wave is
the glyph , which is
hard to decipher, because it appears only on this page and on the
unhelpful page 23. I guess it is intended to refer to the
recipients of the message.

[ Addendum 20151219: Having reviewed page 23, I am still in the
dark.
References to the mass and radius of
suggest that it refers to the recipients’ planet, but references to the mathematics, physics, and biology of
suggests that it refers to the recipients themselves. ]

In the lower-right corner of the page is another cartoon of the RT-70,
this time with a ruler underneath showing its diameter, 70 meters.
Above the cartoon is the power output of the telescope, 150 kilowatts.

The next
article will discuss page 22, shown at
right. (Click to enlarge.) Try to figure it out before then.

These are pages 19–20 of the Cosmic Call
message. An explanation follows.

These two pages are a map of the surface of the Earth.
Every other page in the document is surrounded by a one-pixel-wide frame, to
separate the page from its neighbors, but the two pages that comprise the map
are missing part of their borders to show that the two pages are part of a whole.
Assembled correctly, the two pages are surrounded by a single border.
The matching sides of the map pages have diamond-shaped registration marks to show how to align the two pages.

The map projection used here is R. Buckminster Fuller's Dymaxion
projection, in which the spherical surface of the Earth is first
projected onto a regular icosahedron, which is then unfolded into a
flat net. This offers a good compromise between directional
distortion and size distortion. Each twentieth of the map is
distorted only enough to turn it into a triangle, and the
interruptions between the triangles can be arranged to occur at
uninteresting parts of the map.

Both pages are labeled with
the glyph for “Earth”.
On each page, the land parts of the map are labeled with
and the water parts with
, as on
page 14, since the recipients
wouldn't otherwise be able to tell which was which.

The next
article will discuss page 21, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page depicts the best way to fry eggs. The optimal fried egg is
shown at left. Ha ha, just kidding. The left half of the page
explains cellular respiration. The fried egg is actually a cell, with
a DNA molecule in its nucleus. Will the aliens be familiar enough
with the structure of DNA to recognize that the highly abbreviated
picture of the DNA molecule is related to the nucleobases on the
previous page? Perhaps, if their genetic biochemistry is similar to
ours, but we really have no reason to think that it is.

The illustration of the DNA molecule is subtly wrong. It shows a
symmetric molecule. In reality, one of the two grooves between the
strands is about twice as big as the other, as shown at right.

The top formula says that C6H12O6 and
O2 go into the cell; the bottom formula says that CO2 comes
out. (Energy comes out also; I wonder why this wasn't mentioned.)
The notation for chemical compounds here is different from that used
on
page 14: there, O2 was written as
;
here it is written as
(“2×O”).

The glyph near the left margin does not appear elsewhere,
but I think it is supposed to mean “cell”. Supposing that is correct,
the text at the bottom says that the number of cells in a man or woman
is !!10^{13}!!. The number of cells in a human is not known, except
very approximately, but !!10^{13}!! is probably the right order of
magnitude. (A
2013 paper from Annals of Human Biology estimates
!!3.72\cdot 10^{13}!!.)

Next to the cell is a ruler labeled !!10^{-5}!! meters, which is a
typical size for a eukaryotic cell.

The illustration on the right of the page, annotated with the glyphs
for the four nucleobases from the previous page , depicts the
duplication of genetic material during cellular division. The DNA
molecule splits down the middle like a zipper. The cell then
constructs a new mate for each half of the zipper, and when it
divides, each daughter cell gets one complete zipper.

The next
article will discuss pages 19 and 20, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page depicts the chemical structures of the four
nucleobases that make up
the information-carrying part of the DNA molecule. Clockwise from top
left, they are thymine , adenine , guanine , and cytosine
.

The deoxyribose and phosphate components of the nucleotides, shown at
right, are not depicted. These form the spiral backbone of the DNA
and are crucial to its structure. Will the recipients understand why
the nucleobases are important enough for us to have mentioned them?

The next
article will discuss page 18, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page, about human vital statistics and senses, is in three
sections. The text in the top left explains the population of the
Earth: around 6,000,000,000 people at the time the message was sent.
The three following lines give the life expectancy (70 years), mass
(80 kg), and body temperature (311K) of humans. In each case it is
stated explicitly that the value for men and for women is the same,
which is not really true.
The glyph used for life expectancy is the same one used
to denote the age of the Earth back on page 13 even though the
two notions are not really the same.
And why 311K when the commonly-accepted value is 310K?

The diagram at right attempts to explain the human sense of hearing,
showing a high-frequency wave at top and a low frequency one at
bottom, annotated with the glyph for frequency and the upper
and lower frequency limits of human hearing, 20,000 Hz and 20 Hz
respectively. I found this extremely puzzling the first time I
deciphered the message, so much so that it was one of the few parts of
the document that left me completely mystified, even with the advantage
of knowing already what humans are like. A significant part of the
problem here is that the illustration is just flat out wrong. It
depicts transverse waves:

but sound waves are not transverse, they are compression waves. The
aliens are going to think we don't understand compression waves. (To
see the difference, think of water waves, which are transverse: the
water molecules move up and down—think of a bobbing cork—but the wave itself travels in
a perpendicular direction, not vertically but toward the shore, where it eventually crashes on the
beach. Sound waves are not like this. The air molecules move back
and forth, parallel to the direction the sound is moving.)

I'm not sure what would be better; I tried generating some random
compression waves to fit in the same space.
(I also tried doing a cartoon of a non-random, neatly periodic
compression wave, but I couldn't get anything I thought looked good.)
I think the compression waves are better
in some ways, but perhaps very confusing:

On the one hand, I think they express the intended meaning more
clearly; on the other hand, I think they're too easy to confuse with
glyphs, since they happen to be on almost the same scale. I think
the message might be clearer if a little more space were allotted for
them. Also, they could be annotated with the glyph for pressure , maybe something like this:

This also gets rid of the meaningless double-headed arrow. I'm not
sure I buy the argument that the aliens won't know about arrows; they
may not have arrows but it's hard to imagine they don't know about any sort
of pointy projectile, and of course the whole purpose of a pointy
projectile (the whole point, one might say) is that the point is on
the front end. But the arrows here don't communicate motion or
direction or anything like that; even as a human I'm not sure what
they are supposed to communicate.

The bottom third of the diagram is more sensible. It is a diagram
showing the wavelengths of light to which the human
visual system is most sensitive. The x-axis is labeled with
“wavelength” and the
y-axis with a range from 0 to 1. The three peaks have their centers
at 295 nm (blue), 535 nm (green), and 565 nm (often called “red”, but
actually yellow). These correspond to the three types of cone cells
in the retina, and the existence of three different types is why we
perceive the color space as being three-dimensional. (I discussed
this at greater
length a few
years ago.) Isn't it interesting that the “red” and green
sensitivities are so close together? This is why we have red-green
color blindness.

The next
article will discuss page 17, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page starts a new section of the document, each page headed with the glyph for “biology”
. The illustration
is adapted from the Pioneer
plaque; the relevant
portion is shown below.

Copies of the plaque were placed on the 1972 and 1973 Pioneer
spacecraft. The Pioneer image has been widely discussed and
criticized; see the Wikipedia
article for some of the
history here. The illustration suffers considerably from its
translation to a low-resolution bitmap. The original picture omits
the woman's vulva; the senders have not seen fit to correct this bit
of prudery.

The man and the woman are labeled with the glyphs and , respectively.
The glyph for “people” , which identified
the stick figures on the previous page, is inexplicably omitted here.

The ruler on the right somewhat puzzlingly goes from a bit above the man's toe
to a bit below the top of the woman's head; it does not measure either of the
two figures. It is labeled 1.8 meters, a typical height for men. The
original Pioneer plaque spanned the woman exactly and gave her height
as 168 cm, which is conveniently an integer multiple of the basic measuring unit (21 cm) defined on the plaque.

To prevent the recipients from getting confused about which end of the
body is the top, a parabolic figure (shown here at left), annotated with the glyph
for “acceleration”, shows the direction of gravitational acceleration
as on the previous page.

The next
article will discuss page 16, shown at
right. (Click to enlarge.) Try to figure it out before then.

This is my favorite page: there is a lot of varied information and the
illustration is ingenious. The page heading says to match up with
the corresponding labels on the previous three pages. The page
depicts the overall terrain of the Earth. The main feature is a large
illustration of some mountains (yellow in my highlighted illustration below)
plunging into the sea (blue).

The land part is labeled , the air part , and the water part
. Over on the
left of the land part are little stick figures, labeled people . This is to show
that people live on the land part of the Earth, not under water or in
the air. The stick figures may not be clear to the recipients, but
they are explained in more detail on the next page.

Each of the three main divisions is annotated with its general
chemical composition, with compounds listed in order of prevalence.,
All the chemical element symbols were introduced earlier, on pages
6
and
7:

There are rulers extending upward from the surface of the water to the
height of top of the mountain and downward to the bottom of the ocean.
The height ruler is labeled 8838 meters, which is the height the peak
of Mount Everest, the point highest above sea level. The depth ruler
is labeled 11000 meters, which is the depth of the Challenger Deep in
the Mariana Trench, the deepest part of the ocean. The two rulers
have the correct sizes relative to one another. The human figures at
left are not to scale (they would be about 1.7 miles high), but the next
page will explain how big they really are.

I don't think the message contains anything to tell the recipients
the temperature of the Earth, so it may not be clear that the
hydrosphere is liquid water. But perhaps the wavy line here will
suggest that. The practice of measuring the height of the mountains
and depth of the ocean from the surface may also be suggestive of a
liquid ocean, since it would not otherwise have a flat surface to
provide a global standard.

There is a potential problem with this picture: how will the
recipients know which edge is the top? What if they hold it
upside-down, and think the human figures are pointing down into the
earth, heads downwards?

This problem is solved in a clever way: the dots at the right of the
page depict an object accelerating under the influence of gravity,
falling in a characteristic parabolic path. To make the point clear,
the dots are labeled with the glyph for
acceleration.

Finally, the lower left of the page states the
acceleration due to gravity at the Earth's surface, 9.7978
m/s2. The recipients can calculate this value from the
mass and radius of the Earth given earlier. Linked with the other
appearance of acceleration on the page, this should suggest that the
dots depict an object falling under the influence of gravity toward
the bottom of the page.

The next
article will discuss page 15, shown at
right. (Click to enlarge.) Try to figure it out before then.

There are three diagrams on this page, each depicting something going
around. Although the direction is ambiguous (unless you understand
arrows) it should at least should be clear that all three rotations
are in the same direction. This is all you can reasonably say anyhow,
because the rotations would all appear to be going the other way if
you looked at them from the other side.

The upper left diagram depicts the Earth going around the
Sun and
underneath is a note that says that the time is equal to
315569268 seconds, and is also equal to one year . This defines the
year.

The upper-right
diagram depicts the Moon going around the
Earth ; the
notation says that this takes 2360591 seconds, or around 27⅓
days. This is not the 29½ days that one might naïvely expect, because
it is the sidereal month rather than the synodic month. Suppose the
phase of the Moon is new, so that the Moon lies exactly between the
Earth and the Sun. 27⅓ days later the Moon has made a complete trip around
the Earth, but because the Earth has moved, the Moon is not yet again
on the line between the Earth and the Sun; the line is in a different
direction. The Earth has moved about !!\frac1{13}!! of the way around
the sun, so it takes about another !!\frac1{12}\cdot 27\frac13!! days
before the moon is again between Earth and Sun and so a total of about
29½ days between new moons.

The lower-right diagram depicts the rotation of the Earth, giving a time of
86163 seconds for the day. Again, this is not the 86400 seconds one
would expect, because it is the sidereal day rather than the solar
day; the issue is the same as in the previous paragraph.

None of the three circles appears to be circular. The first one is
nearly circular, but it looks worse than it is because the Sun has been
placed off-center. The curve representing the Moon's orbit is
decidedly noncircular. This is reasonable, because the Moon's orbit
is elliptical to approximately the same degree. In the third diagram,
the curve is intended to represent the surface of the Earth, so
its eccentricity is indefensible. The ellipse is not the same as the
one used for the Moon's orbit, so it wasn't just a copying mistake.

The last two lines state that the ages of the Sun and the Earth are each
4550000000 years. This is the first appearance of the glyph
for “age”.

The next
article will discuss page 14, shown at
right. (Click to enlarge.) Try to figure it out before then.

Page 12, begins a new section of the document, with pages headed with
the glyph
“Earth”, describing the Earth and its environs. This will help the recipients locate our planet, should they come to visit.

The top of the page
has a diagram of the Earth -Moon system and a ruler
labeled with the distance between them, !!3844\cdot 10^5!! meters. Since the distance between Earth and Moon varies (the orbit is elliptical) an average value is given.

The glyph used here for distance, , is different
from the one defined on page 9 . Neither
appears elsewhere in the message, so this is probably a mistake.

The following five lines give the mass
and radius of the Moon and the Earth, and also the distance from the Earth to the Sun. The latter would have been better on the previous page, which discussed the solar system, but was omitted from there for some reason.

The next
article will discuss page 13, shown at
right. (Click to enlarge.) Try to figure it out before then.

Page 11, headed with the glyph for “physics” is evidently a
chart of the solar system, with the Sun at left. The Earth
is also
labeled, as is Jupiter , the planet most
likely to be visible to the recipients. The “Jupiter” glyph does not appear again.
Pluto is included, as it was still considered a planet in 1999. (Pluto's
status as only one of many similar trans-Neptunian objects was not
well appreciated in 1999 when the message was composed, the second TNO
having only been discovered in 1992.)
To the extent permitted by the low resolution of the image,
The diameters of the planets and
the Sun are to scale,
but not their relative positions; the page is
much too small for that. Saturn's rings are not shown, as they are in
the Pioneer plaque; by
this time it was appreciated that ring systems may be common around large
planets.

The masses
and radii
of Jupiter and the Sun are given, Jupiter above
the illustration and the Sun below. The (external) temperature of the Sun
is also given, 5763 kelvins. This should be visible to the aliens
because the Sun is a blackbody emitter and the spectrum of blackbody
radiation is a clear indicator of its temperature. This data should
allow the aliens to locate us, should they be so inclined: they know
which way the message came from, and can look for a star with the
right size and temperature in that direction. When they get closer,
Jupiter and the sizes of the planets will provide a confirmation that
they are in the right place. Later pages explain that we live on the
Earth, so the aliens will know where to point their fusion cannon in
order to obliterate our planet.

The next
article will discuss page 12, shown at
right. (Click to enlarge.) Try to figure it out before then.

The top half of this page is a table of melting points (on the left)
and boiling points (on the right) for various substances: hydrogen
, carbon
, sulfur
, zinc , silver , and gold . The temperatures
are given in kelvins .

The boiling points depend on pressure, so there is a notation at the
bottom of the list that the pressure should be
101300 pascals . This is one standard atmosphere, so it may tell the aliens a
little more about our planet.

To help calibrate the kelvins, the bottom of the page is a chart
of the temperature
increase of water
, showing how the temperature stops increasing at the
melting point (273K) and the boiling point (373K). This introduces the
glyph for temperature , which will recur later.

There are two
regrettable things about this chart. One is that the horizontal axis
is labeled “time”
. Why is the
temperature of the water increasing with time? It should be energy.

But a more serious complaint, I think, it that this is the wrong
chart. It depends crucially on the (Earth-)standard atmospheric pressure,
with which the recipients may not be familiar. And the kelvin is not defined
in terms of standard pressure anyway. It is defined in terms of the
triple point of water, the unique, universal temperature and pressure at which
all three states of water can coexist. Why not a temperature and
pressure chart with the triple point labeled? This is something one
might more reasonably expect the aliens to have studied.

The next
article will discuss page 11, shown at
right. (Click to enlarge.) Try to figure it out before then.

The previous two pages defined fundamental units of mass, distance,
and time. This page adds some derived notions: force, energy, pressure, and
power; velocity and acceleration.

The first four lines are in two parts each. The left part defines an
abstract quantity like force or energy; the right part defines a unit
of that quantity like newtons or joules. For example, the second line defines energy and units of energy. The left side says that energy
is equal to mass
times distance
squared divided by time squared. (This is the first appearance of the glyphs for distance and time.)
The right side says that a joule
of energy is !!1\frac{\mathrm{kg}\,\mathrm{m}^2}{\mathrm{s}^2}!!.

The two following lines define the abstract concepts of velocity (which has
appeared before in connection with the speed of light) and
acceleration (which is
new). There are no units given for these.

Concept

force

energy

pressure

power

velocity

acceleration

Unit

newton

joule

pascal

watt

(none)

(none)

The newton and the joule won't appear again. Force won't appear again
except on the last page, which asks “force WTF?”.

The final part of the page is the most interesting. It mentions the
Planck constant !!h!!
(not the related !!\hbar!!) which is
!!6.6260755\cdot 10^{{}^-34}!! joule-seconds and the universal gravitation constant !!G!!
which is !!6.67259\cdot 10^{{}^-11}\; \mathrm{m}^3\;
\mathrm{kg}^{{}^-1}\; \mathrm{s}^{{}^-2}!!.

It's quite possible that the recipients won't recognize the gravitational
constant, which is tricky to measure directly. Newton's law of
universal gravitation was known on Earth for hundreds of years before
the value of !!G!! was worked out. But if the recipients don't know
it, they will be able to work it out from the later statements about
the mass and radius of the Earth and the gravitational force at its
surface, This would in turn allow them to calculate the mass of their
own planet.

Note that on this page some
inverse units are written with a division sign and some with a
negative exponent. Will this inconsistency puzzle the aliens? My
coworker Jeff Ober suggests that it communicates the important
personal information that humans are confused and inconsistent.

The next
article will discuss page 10, shown at
right. (Click to enlarge.) Try to figure it out before then.

The main feature of this page is a diagram of the electron energy
levels for a hydrogen atom, annotated at the top with the glyph for
hydrogen
and at the bottom with the glyph for energy . The four lowest
levels are shown, with the lowest level (the ground state) at the
bottom. Above these is a thicker bar representing the way the higher
energy levels all pile up into a smear. If the bottom level is at 0
and the smear is at 1, then the three intermediate levels are
shown at approximately their exact values of !!\frac34, \frac89, !! and !!\frac{15}{16}!!; these are
given by the Rydberg
formula.

The aliens should be familiar with hydrogen.
Normally a hydrogen atom's sole electron is in the ground state. If a
photon couples with the electron, say because starlight is falling on
the atom, or someone has applied an electric current to it, the
electron may jump up to a higher quantum state. (This is the only
correct use of the phrase "quantum leap".) When it drops back down,
it will emit a photon. But the energy of the emitted photon must be
one of a few particular values, corresponding to the difference
between the old and the new energy level; the electron never drops
down partway to the next level. An incandescent cloud of hydrogen gas
will not typically glow in every possible color; it will emit light in
only a few particular, characteristic wavelengths, and these colors
can be separated with a spectroscope. The wavelengths of these
characteristic colors of light, visible everywhere the message is
likely to reach, provide a basis for defining the meter.

For example, the second pair of numbers labels the transition from the
!!n=3!! to the !!n=2!! state, and an electron transiting between these
two states will always emit a photon with a wavelength of 656.2852
nanometers and a frequency of 456.8021 terahertz, and these are the
two numbers in the pair. The product of the two numbers in each pair
is a constant, which should further confirm to the recipients that
they have the right interpretation. The constant product is close to
299792458, which is the speed of light in meters per second. This
defines four glyphs, for wavelength and frequency, and meters and
hertz.

wavelength

frequency

meters

hertz

seconds

The following item

defines the second as the inverse of
the hertz (written as !!\mathrm{Hz}^{{}^-1}!! and as
!!1\div\mathrm{Hz}!!).

Finally, the speed of light is given, first
theoretically, as the product of wavelength and frequency and then
numerically, as 299792458 meters per second .

The next
article will discuss page 9, shown at
right. (Click to enlarge.)
Try to figure it out before then.

The top section of the page continues the table of chemical elements
from the previous page, giving the number of protons and neutrons.
For example, gold
is described as having 79 protons and 117 neutrons
.

Sulfur

Zinc

Argon

Silver

Gold

Uranium

Copernicium

Copernicium, element 112, had been discovered but not named at the
time the document was written.

There is a major error here.
Uranium
is given as having 92 protons and 116
neutrons. There is no such substance. It should have said 146
neutrons.

I sometimes imagine the aliens, having received the
message, come to visit us. “We weren't going to bother,” they say,
“but we had to know about the uranium-208.” And then we will have to
tell them that we messed up. Ouch. (It could be an error for lead-208
or bismuth-208 instead; one can't be sure because the glyph does not
appear elsewhere in the document.)

I'd been planning to write that paragraph about uranium-208 for more
than ten years, but it wasn't until just now that I realized there is
a much more serious mistake two lines down, so that the uranium is no longer
the most serious error that I know of in the entire document. The
line after the table of elements says that the mass of a carbon atom is the mass of six
protons plus the mass of six neutrons plus “energy”, , by which I think
they mean the binding energy in the nucleus. This is the first
appearance of the glyph for energy, which will recur later. And then
the following line commits a really horrible boner, one that has the
potential to spoil the whole message.

With the mass of the carbon nucleus pinned down, the authors want to
define the kilogram : the document
says that 12 kilograms is the mass of !!6022137\cdot 10^{19} !!
carbon-12 atoms. That !!6022137\cdot 10^{19} !! is Avogadro's number.
Except it's not. Avogadro's number is usually given as
!!6.022137\cdot 10^{23} !!, and this number is 100 times that big.
But it should be 1000 times that big.

Normally, one would say that !!6.022137\cdot 10^{23} !! carbon atoms
mass 12 grams, but there are two confusing factors here. One is that
the authors have written !!6022137!! instead of !!6.022137!! and the
other is that they are defining 12 kilograms instead of 12 grams.
But it should be that !!6.022137\cdot 10^{23} = 6022137\cdot 10^{17}!!
atoms is 12 grams so that !!6022137\cdot 10^{20}!! atoms is 12 kilograms,
and the number written is instead !!6022137 10^{19}!! atoms, making
the kilogram 90% smaller than it should have been.

It's possible that the aliens can figure this out, because it is
detectably inconsistent with the following statements about the masses
of the fundamental particles in kilograms. But it may not be clear to
the recipients which of the two definitions of the kilogram is the
correct one. Especially given the—I really hate to report this—the
typo in the second statement.

The three following lines give the masses of the proton, neutron, and
electron in kilograms. These are all more or less correct (although
the book values have changed since the message were written) and I
think the value for the neutron has a typo; it says !!1.6739286\cdot
10^{{}^-34}!! kg but it probably should have been !!1.67{\mathit
4}9286\cdot 10^{{}^-34}!! kg which would agree with the current book
value of !!1.674927351\cdot 10^{{}^-34}!! kg.

Since we're going over the errors on this page, here is yet another
oddity. The number of neutrons in a gold atom
is given at the top of
the page as 117. Unlike uranium-208., the isotope gold-196 actually exists. But it is radioactive,
breaking down into platinum or mercury after about a week. One would
expect the listing to be for gold-197 instead, which is the only
stable isotope and so is the only isotope occurring in naturally-found
gold. (Thanks to Peter Annema for bringing this to my attention.)
A similar oddity occurs in the listing for zinc
:
zinc-65 is given instead of the stable zinc-64 or zinc-66.
The
other isotopes listed here (sulfur-32, argon-40, silver-107) are more
plausible.

The next
article will discuss page 8, shown at
right. (Click to enlarge.) Try to figure it out before then.

Page 6 discusses fundamental particles of matter, the structure of the
hydrogen and helium atoms, and defines glyphs for the most important
chemical elements.

Depicted at top left is the hydrogen atom, with a proton in the center and
an electron circulating
around the outside. This diagram is equated to the glyph for hydrogen.

The diagram for helium
is similar but has two electrons, and its nucleus has two protons and
also two
neutrons.

Proton

Neutron

Electron

The illustrations may puzzle the aliens, depending on how they think of
atoms. (Feynman once said that this idea of atoms as little solar
systems, with the electrons traveling around the nucleus like planets,
was a hundred years old and out of date.) But the accompanying mass
and charge data should help clear things up. The first formula says

!!M_p = 1836\cdot M_e!!

the mass of the proton is 1836
times the mass of the electron, and that 1836, independent of the
units used and believed to be a universal and fundamental constant,
ought to be a dead giveaway about what is being discussed here.

If you want to communicate fundamental constants, you have a bit of a
problem. You can't tell the aliens that the speed of light is
!!1.8\cdot10^{12}!! furlongs per fortnight without first explaining
furlongs and fortnights (as is actually done on a later page). But the
proton-electron mass ratio is dimensionless; it's 1836 in every
system of units. (Although the value is actually known to be
1836.15267; I don't know why a more accurate value wasn't given.)

This is the first use of subscripts in the document. It also takes
care of introducing the symbol for mass. The
following formula does the same for charge : !!Q_p = -Q_e!!.

The next two formulas, accompanying the illustration of the helium
atom, describe the mass (1.00138 protons) and charge (zero) of the
neutron. I wonder why the authors went for the number 1.00138 here
instead of writing the neutron-electron mass ratio of 1838 for
consistency with the previous ratio. I also worry that this won't be
enough for the aliens to be sure about the meaning of . The 1836 is as
clear as anything can be, but the 0 and -1 of the corresponding charge
ratios could in principle be a lot of other things. Will the context
be enough to make clear what is being discussed? I suppose it has to; charge, unlike mass, comes in discrete units and there is nothing like the 1836.

The second half of the page reiterates the symbols for hydrogen and
helium and defines symbols for eight other chemical elements. Some of
these appear in organic compounds that will be discussed later; others
are important constituents of the Earth. It also
introduces symbol for “union” or “and”: . For example,
sodium is described as having 11 protons and 12 neutrons.

Hydrogen

Helium

Carbon

Nitrogen

Oxygen

Aluminium

Silicon

Iron

Sodium

Chlorine

Most of these new glyphs are not especially mnemonic, except for hydrogen—and
aluminium, which is spectacular.

The blog is going on hiatus until early September. When it returns, the next
article will discuss page 7, shown at
right. It has three errors. Can you find them? (Click to enlarge.)

Page 5 discusses two basic notions of geometry. The top half concerns
circles and introduces !!\pi!!. There is a large circle with its
radius labeled :

The outer
diameter is then which is !!2\cdot r!!.

The perimeter
is twice
times the radius
,
and the area
is
times the square of the radius . What is ? It's !!\pi!! of
course, as the next line explains, giving !!\pi =
3.1415926545697932…365698614212904!!, which gives enough digits on the
front to make clear what is being communicated. The trailing digits
are around the 51 billionth places and communicate part of the state
of our knowledge of !!\pi!!. I almost wish the authors had included
a sequence of fifteen random digits at this point, just to keep the aliens
wondering.

The bottom half of the page is about the pythagorean theorem. Here
there's a rather strange feature. Instead of using the three variables
from the previous page,
,
the authors changed the second one and used
instead. This new glyph does not
appear anywhere else. A mistake, or did they do it on purpose?

In any case, the pythagorean formula is repeated twice, once with
exponents and once without, as both !!z^2=x^2+b^2!! and !!z\cdot z =
x\cdot x + b\cdot b!!. I think they threw this in just in case the
exponentiation on the previous pages wasn't sufficiently clear. I
don't know why the authors chose to
use an isosceles right triangle; why not a 3–4–5 or some other scalene
triangle, for maximum generality? (What if the aliens think we think
the pythagorean theorem applies only for isosceles triangles?) But
perhaps they were worried about accurately representing any funny
angles on their pixel grid. I wanted to see if it would fit, and it does. You have to make the diagram smaller, but I think it's still clear:

(I made it smaller than it needed to be and then didn't want to redo it.)

I hope this section will be sufficiently
unmistakable that the aliens will see past the oddities.

The next
article will discuss page 6, shown at
right. (Click to enlarge.) Try to figure it out before then.

And the equal sign .
Page 2 explained the four basic arithmetic operations and some associated notions:

addition

subtraction

multiplication

division

negation

ellipsis (…)

decimalpoint

indeterminate

This page, headed with the glyph for “mathematics” , describes the solution of simple algebraic equations and defines glyphs for three variables, which we may as well call !!x,y,!! and !!z!!:

x

y

z

Each equation is introduced by the locution which means “solve
for !!x!!”. This somewhat peculiar “solve” glyph will not appear
again until page 23.

For example the second equation is !!x+4=10!!:

Solve for !!x!!: !!x+4=10!!

The solution, 6, is given over on the right:

!!x=6!!

After the fourth line, the equations to be solved change from simple
numerical equations in one variable to more abstract algebraic
relations between three variables. For example, if

Solve for !!x!!: !!x\cdot y=z!!

then

!!x=z\div y!!.

The next-to-last line uses a decimal fraction in the exponent,
!!0.5!!:
. On the previous page, the rational fraction !!1\div 2!! was
used. Had the same style been followed, it would have looked like this:
.

Finally, the last line defines !!x=y^3!! and then, instead of an
algebraic solution, gives a graph of the resulting relation, with axes
labeled. The scale on the axes is not the same; the !!x!!-coordinate
increases from 0 to 20 pixels, but the !!y!!-coordinate increases from 0 to 8000 pixels because !!20^3 = 8000!!. If
axes were to the same scale, the curve would go up by 8,000 pixels.
Notice that the curve does not peek above the !!x!!-axis until around
!!x=8, y=512!! or so. The authors could have stated that this was the graph
of !!y=x^3\div 400!!, but chose not to.

I also wonder what the aliens will make of the arrows on the axes. I
think the authors want to show that our coordinates increase going up
and to the left, but this seems like a strange and opaque way to do
that. A better choice would have been to use a function with an
asymmetric graph, such as !!y=2^x!!.

(After I wrote that I learned that similar concerns were voiced about
the use of a directional arrow in the Pioneer
plaque.

(Wikipedia
says: “An article in Scientific American criticized the use of an
arrow because arrows are an artifact of hunter-gatherer societies like
those on Earth; finders with a different cultural heritage may find
the arrow symbol meaningless.”)

The next
article will discuss page 5, shown at
right. (Click to enlarge.) Try to figure it out before then.

And the equal sign .
Page 2 explained the four basic arithmetic operations and some associated notions:

addition

subtraction

multiplication

division

negation

ellipsis (…)

decimalpoint

indeterminate

This page, headed with the glyph for “mathematics” , explains
notations for exponentiation and scientific notation. (This notation
was first used on page 1 in the mersenne prime .)

Exponentiation could be represented by an operator, but instead the
authors have chosen to represent it by a superscripted position on the
page, as is done in conventional mathematical notation. This saves
space.

The top section of the page has small examples of exponentiation, including for example !!5^3=125!!:

!!5^3=125!!

There is a section that follows with powers of 10: !!10^1=10,
10^2=100, 10^3=1000, !! and more interestingly !!10^{{}^-2} = 0.01!!:

!!10^{{}^-2} = 0.01!!

This is a lead-in to the next section, which expresses various
quantities in scientific notation, which will recur frequently later
on. For example, !!0.045!! can be written as !!45\times 10^{{}^-2}!!:

!!45\times10^{{}^-2} = 0.45!!

Finally, there is an offhand remark about the approximate value of the square root of 2:

!!2^{1\div 2} = 1.41421356…!!

The next
article will discuss page 4, shown at
right. (Click to enlarge.) Try to figure it out before then.

This page, headed with the glyph for “mathematics” , explains the arithmetic operations on numbers.

The page is in five sections, three on top and two below.
The first four sections explain
addition ,
subtraction ,
multiplication , and
division . Each is explained with a series of five typical arithmetic equalities. For example, !!4\times 3= 12!!:

The subtraction sign actually appeared back on page 1
in the Mersenne prime !!2^{3021377}-1!! .

The negative sign is introduced in
connection with subtraction, since !!1-2={}^-1!!:

Note that the negative-number sign is not the same as the
subtraction sign.

The decimal point is
introduced in connection with division. For example, !!3\div 2 =
1.5!!:

There is also an attempt to divide by zero:

It's not clear what the authors mean by this; the mysterious glyph
does
not appear anywhere else in the document. What did they think it
meant? Infinity? Indeterminate? Well, I found out later they
published a cheat sheet, which assigns the meaning “undetermined” to
this glyph. Not a great choice, in my opinion, because !!1÷0!! is not
numerically equal to anything.

For some reason, perhaps because of space limitations, the authors
have stuck the equation !!0-1 = {}^-1!! at the bottom of the division
section.

This page, headed with the glyph for “mathematics” , explains
the numeral symbols that will be used throughout the rest of the
document. I should warn you that these first few pages are a little
dull, establishing basic mathematical notions. The good stuff comes a
little later.

The page is in three sections. The first section explains the
individual digit symbols. A typical portion looks like this:

•••• ••• = 0111 = 7

Here the number 7 is written in three ways: first, as seven dots,
probably unmistakable. Second, as a 4-bit binary number, using the
same bit symbols that are used in the page numbers. The three forms
are separated by the glyph , which means “equals”.
The ten digits, in order from 0 to 9, are represented by the glyphs

0

1

2

3

4

5

6

7

8

9

The authors did a great job selecting glyphs that resemble the
numerals they represent. All have some resemblance except for 4,
which has 4 horizontal strokes. Watch out for 4; it's easy to confuse with 3.

The second section serves two purposes. It confirms the meaning of
the ten digits, and it also informs the aliens that the rest of the
message will write numerals in base ten. For example, the number 14:

••••• ••••• •••• = 14

Again, there are 14 dots, an equal sign, and the numeral 14, this time
written with the two glyphs (1) and (4). The base-2
version is omitted this time, to save space. The aliens know from this
that we are using base 10; had it been, say, base 8, the glyphs would
have been .

People often ask why the numbers are written in base 10, rather than
say in base 2. One good answer is: why not? We write numbers in base
10; is there a reason to hide that from the aliens? The whole point
of the message is to tell the aliens a little bit about ourselves, so
why disguise the fact that we use base-10 numerals? Another reason is
that base-10 numbers are easier to proofread for the humans sending
the message.

The third section of the page is a list of prime numbers from 2 to 89:

67, 71, 73, 79, 83

and finally the number !!2^{3021377}-1!!

,!!2^{3021377}-1!!

which was the
largest prime number known to humans at the time. (The minus sign and
exponentiation notation are explained on later pages.) Why?
Again. to tell the aliens about ourselves: here's a glimpse of the
limits of our mathematical knowledge.

I often wonder what the aliens
will think of the !!2^{3021377}-1!!. Will they laugh at how cute we
are, boasting about the sweet little prime number we found? Or will
they be astounded and wonder why we think we know that such a big
number is prime?

The next
article, to appear 2015-08-12, will
discuss page 2, shown at right. (Click to enlarge.) Try to figure it
out before then.

(At left is page 1 of the Cosmic Call
message. For an enlarged version of the image, click it.)

First, some notes about the general format of each page. The Cosmic
Call message was structured as 23 pages, each a 127×127 bitmap.
The entire message was therefore 127×127×23 bits, and
this would hopefully be suggestive to the aliens: they could try
decoding it as 127 pages of 127×23-bit images, which would
produce garbage, or as 23 pages of 127×127-bit images, which is
correct. Or they might try decoding it as a single 127×2921-bit
image, which would also work. But the choices are quite limited and
it shouldn't take long to figure out which one makes sense.

To
assist in the framing, each page of the message is surrounded by a
border of black pixels and then a second smaller border of white
pixels. If the recipient misinterpreted the framing of the bit sequence,
say by supposing that the message was made of lines of 23 pixels, it
would be immediately apparent that something was wrong, as at right.
At the very least the regular appearance of the black border pixels
every 127 positions, and the fact that the message began with 128
black pixels, would suggest that there was something significant about
that number. If the aliens fourier-transform the message, there
should be a nice big spike at the 127 mark.

Most of the message is encoded as a series of 5×7-pixel glyphs. The
glyphs were generated at random and then subject to later filtering:
candidate glyphs were discarded if they don't differ from previous
glyphs in enough bit positions. This is to help the recipients
reconstruct the glyphs if some of the bits are corrupted in
transmission, as is likely.

The experimenters then eyeballed the glyphs and tried to match glyphs
with their meanings in a way that would be easy for humans to
remember, to aid in proofreading. For example, the glyph they chose
to represent the digit 7 was .

People frequently ask why the message uses strange glyphs instead of
standard hindu-arabic numerals. This is explained by the need to have
the glyphs be as different as possible. Communication with other
stars is very lossy. Imagine trying to see a tiny flickering light
against the background of a star at a distance of several light years.
In between you and the flickering light are variable dust and gas
clouds. Many of the pixels are likely to be corrupted in transmission.
The message needs to survive this corruption. So glyphs are 35 bits
each. Each one differs from the other glyphs in many positions,
whereas a change of only a few pixels could change a standard 6 into
an 8 or vice versa. A glyph is spread across multiple lines of the
image, which makes it more resistant to burst errors: even if an
entire line of pixels is destroyed in transit, no entire glyph will be
lost.

At the top left and top right of each page are page numbers. For
example, page number 1: The page numbers
are written in binary, most significant bit first, with
representing a 1 bit and
representing a 0 bit. These bit
shapes were chosen to be resistant to data corruption; you can change
any 4 of the 9 pixels in either shape and the recipient can still
recover the entire bit unambiguously. There is an ambiguity about
whether the numerals are written right to left or left to right—is
the number 1
or the number 16?—but the aliens should be able to figure it out by
comparing page numbers on consecutive pages; this in turn will help
them when time comes for them to figure out the digit symbols.

Every page has a topic header, in this case , which roughly means
“mathematics”. The topics of the following pages are something like:

1–5 Mathematics

6–11,21 Physics

12–14,19–20 The Earth

15–18 Human anatomy and biochemistry

22 Cosmology

23 Questions

In the next article I'll explain the contents of page 1. Each
following article will appear two or three days later and will explain
another page.

To follow the whole series of articles in your feed reader, subscribe to:
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The message images themselves are extremely compelling. I saw the
first one in the book Beyond
Contact by Brian McConnell:

I didn't think much of the rest of the book, but the image was
arresting. After staring at it for a while, and convincing myself I
understood the basic idea, I found the full set of images on Mike Matessa's web
site, printed them out, and spent a happy couple of hours at the kitchen table
deciphering them.

Sometimes when I gave conference talks, I would put this image on the
screen during break, to give people something to think about before
the class started up again. I like to say that it's fun to see if
you're as smart as an alien, or at least if as smart as the Canadian
astrophysicists thought the aliens would be.

I invite you to try to understand what is going on in the first image,
above. In a day or two I will post a full explanation, along with the
second image. Over the next few weeks I hope to write a series of blog
articles about the 23 pages, explaining the details of each.